Well, I've never heard nor seen these "Eidolons" before, but we did have to survive each summon's onslaught. Not a simple task, I assure you.

[Granted, it helped to apply elemental resistances. Accessories, support abilities... Anything to get the job done.]

So these creatures undergo metamorphosis through battle? My, that's certainly nothing like the monsters in Luxendarc nor summons. The closest thing I can think of are the job asterisks-- become proficient enough, and you'll obtain new skills that can turn the tide in any fight.